Theory of attenuation and finite propagation speed in viscoelastic media

TL;DR

This paper investigates the relationship between attenuation, dispersion, and finite propagation speed in viscoelastic media, establishing conditions under which finite speed is maintained and showing superlinear attenuation is incompatible with it.

Contribution

It provides a theoretical framework linking positive relaxation spectra with finite propagation speed and characterizes the high-frequency behavior of attenuation functions.

Findings

Positive relaxation spectrum leads to sublinear high-frequency growth of attenuation.

Finite propagation speed constrains the high-frequency behavior of attenuation functions.

Superlinear power law attenuation is incompatible with finite speed and positive relaxation spectrum.

Abstract

It is shown that the dispersion and attenuation functions in a linear viscoelastic medium with a positive relaxation spectrum can be expressed in terms of a positive measure. Both functions have a sublinear growth rate at very high frequencies. In the case of power law attenuation positive relaxation spectrum ensures finite propagation speed. For more general attenuation functions the requirement of finite propagation speed imposes a more stringent condition on the high-frequency behavior of attenuation. It is demonstrated that superlinear power law frequency dependence of attenuation is incompatible with finite speed of propagation and with the assumption of positive relaxation spectrum.